Be careful with whether you believe you must have an animal spirit guide, or whether you believe you do. There are a lot of New Age sources which insist everyone must have one, as a necessity. This is not true.
Some people do not believe in spirit guides, or even spirits for that matter. But that is a bit of a digression.
Spirit guides are met through meditation -- not forced meditation, mind you. I find it quite irritating when I find guided meditations or instructions on exactly what to visualize, to attempt to force an encounter with a spirit guide. It is far too easy in these cases for one's own desires and imagination to interfere. These guided meditations are not bad; just be careful with them.
They can also be met though dreams, astral projection, and other forms of journeying (whatever your tradition may practice; I would fear trying to make a list, as it would be both incomplete and inaccurate to at least some traditions).
In some traditions, everyone has a guide or guides, which can be sought and found. Some traditions teach that a guide will present itself when ready. Some traditions hold that guides may change as a person develops.
If your guide chooses to express itself to you in an animal form, then it may be your animal spirit guide.
Also consider whether you have an affinity for a specific animal for no explainable reason. These may be signs.
But the answer to your question is an inward quest, and not an outward one.
I suppose the best answer to connecting with any spiritual guide/guardian is threefold.
First, study and understand what it is you want to connect with, so you can be precise in what you are looking for. Take some time to be Self-reflective and figure out what you are looking for. It helps to examine what you are hoping to gain from forming that connection. Are you looking for a protective spirit, or an entity to act as a guide/teacher? Are you wanting help in learning about yourself, or about a particular practice? Are you looking for a helpful entity to assist you in your practices by lending its expertise or energy to your workings?
Once you have some understanding of your goals and how you hope to gain from the connection, compare that with information about the different kinds of entities out there and the roles they fulfill. For example spirit guides don't necessarily have to be animals, and act as teachers to help one find themselves and their best path. Guardian spirits are more protectors than teachers and work to soften the blow of things as you learn. Animal spirits are more symbolic in representation and act as a mirror. They tend to operate by helping you observe yourself and understand places where you might have aptitude or have things come naturally, along with other aspects where you might find more challenge or might need to work harder on to improve yourself.
For example finding you connect with an eagle as a spirit animal might reflect on your personality in having a talent for seeing beneath the surface of situations, being able to dive in and pluck out the important details. It might mean having a sense of strength or moral character, clarity of sight or foresight. But also a tendency to escape by flying above/away from challenging situations rather than facing them. Or a tendency to lose your grounding. or an unfortunate habit of being a bit -too- direct and sacrificing tact when it might be needed.
Second, find a personal and formal way of recognizing and expressing your intent. You want to consciously create a consistent, enduring focus of thought towards the goals you want to achieve, and the intent to connect with a being that will best assist you in reaching those goals. Even if that goal is a general idea like 'become more spiritual' or 'connect with magical understanding' or 'understand my path in this life'. It helps in being specific to what you want help with, which will help attract the being best suited to help you with it.
For some, this formal process might be as simple as a daily meditation. For others it might be a more structured process such as a spell or rite. Base this off of what you already do or connect with. If you are studying or practicing a magical path, feel free to hunt down or build a spell that suits your needs. If you are more of an energy worker or intuitive, then hunt down a guided meditation. And of course if you have friends who have a personal practice then you can always ask one of them to take you through a ritual or other process to help you out as well. For the latter it is always best if you can be present and take part in the working in most practices. A few shamanistic paths involve having a master or elder do such a thing for you though, once you have reached a certain point of initiation.
The third part is learning how to be still and listen, and to manage expectations as much as you can. Meditation is a pretty universal tool for this, as the core of that practice is focused on being still and relaxed, and finding awareness in the present moment. IE; being here and now. But the entity you call for might make itself known to you in other ways as well that are hard to predict. Such as in dreams, or you might start noticing animals or other figures consistently appearing in unusual significant places like in the clouds, in books, as gifts, etc. So practice keeping yourself consciously aware of the different hints and signals that might be coming your way.
Look for repeating circumstances, and anything that might seem significant you should make an effort to investigate. Like with most spiritual matters the answer that comes is usually the one that you most need, more than the one you think you want.
For example I wasn't very far along in my own learning when I started trying to connect with my guide. I was fortunate enough to know a practicing wiccan who took me through a guide calling spell, but it still took a year almost to the day for us to finally connect because i was very bad at paying attention.
Every time I sat and wondered or tried to be still and listen, I kept focusing on ideas of what my guide might be or how he/she would make itself known. I kept thinking about Dolphins, or jaguars, or eagles or hawks or... well... anything really. But because I kept looking -for- things, whenever my guide tried to make himself known I would miss it because it didn't match expectations. I would be looking for jaguar, and find a book on dragons. I would be thinking about Dolphins, and get a gift of a dragon statue. I would be looking for eagles, and see dragons in the clouds. I would meditate and ask for my guide to speak to me, and assume the reply I heard was made-up.
In the end, it took a combination of the right timing, a moment of open thought, some inspired automatic writing, and my guide all but beating down the damn door in a dream to finally get my attention strongly enough for me to sit up and take notice, and entertain the possibility that my guide might be a dragon instead of some earthly animal or entity.